Bottom Line:
The data topological analysis will select global features, reducible neither to a mere subgraph nor to a metric or vector space.How the immune system reacts, how it evolves, how it responds to stimuli is the result of an interaction that took place among many entities constrained in specific configurations which are relational.Within this metaphor, the proposed method turns out to be a global topological application of the S[B] paradigm for modeling complex systems.

Affiliation: School of Science and Technology, University of Camerino, Camerino, Italy.

ABSTRACT

In order to define a new method for analyzing the immune system within the realm of Big Data, we bear on the metaphor provided by an extension of Parisi's model, based on a mean field approach. The novelty is the multilinearity of the couplings in the configurational variables. This peculiarity allows us to compare the partition function [Formula: see text] with a particular functor of topological field theory-the generating function of the Betti numbers of the state manifold of the system-which contains the same global information of the system configurations and of the data set representing them. The comparison between the Betti numbers of the model and the real Betti numbers obtained from the topological analysis of phenomenological data, is expected to discover hidden n-ary relations among idiotypes and anti-idiotypes. The data topological analysis will select global features, reducible neither to a mere subgraph nor to a metric or vector space. How the immune system reacts, how it evolves, how it responds to stimuli is the result of an interaction that took place among many entities constrained in specific configurations which are relational. Within this metaphor, the proposed method turns out to be a global topological application of the S[B] paradigm for modeling complex systems.

Fig4: Simplices in . 0-simplex is point or vertex, 1-simplex is an edge, 2-simplex is a triangle, and 3-simplex is a thetrahedron

Mentions:
In this appendix we describe a general approach that allows to extract global topological information from a space of data. It is based on three basic steps: (i) The interpretation of the huge collection of that constitutes the space of data; this is achieved by resorting to a family of simplicial complexes (Fig. 4), parametrized by some suitably chosen ‘proximity parameter’ (Fig. 5). This operation converts the data set into a global topological object. In order to fully exploit the advantages of topology, the choice of such parameter should be metric independent. In our context it measures the expression of a possible . (ii) The reduction of noise, affecting the data space, as the result of the parametrized persistent homology. (iii) The encoding of the data set persistent homology in the form of a parameterized version of topological invariants, in particular Betti numbers, i.e. the invariant dimensions of the homology groups. These three steps provide an exhaustive knowledge of the global features of the space of data, even though such a space is neither a metric space nor a vector space, as other approaches require (Carlsson 2009).Fig. 4

Fig4: Simplices in . 0-simplex is point or vertex, 1-simplex is an edge, 2-simplex is a triangle, and 3-simplex is a thetrahedron

Mentions:
In this appendix we describe a general approach that allows to extract global topological information from a space of data. It is based on three basic steps: (i) The interpretation of the huge collection of that constitutes the space of data; this is achieved by resorting to a family of simplicial complexes (Fig. 4), parametrized by some suitably chosen ‘proximity parameter’ (Fig. 5). This operation converts the data set into a global topological object. In order to fully exploit the advantages of topology, the choice of such parameter should be metric independent. In our context it measures the expression of a possible . (ii) The reduction of noise, affecting the data space, as the result of the parametrized persistent homology. (iii) The encoding of the data set persistent homology in the form of a parameterized version of topological invariants, in particular Betti numbers, i.e. the invariant dimensions of the homology groups. These three steps provide an exhaustive knowledge of the global features of the space of data, even though such a space is neither a metric space nor a vector space, as other approaches require (Carlsson 2009).Fig. 4

Bottom Line:
The data topological analysis will select global features, reducible neither to a mere subgraph nor to a metric or vector space.How the immune system reacts, how it evolves, how it responds to stimuli is the result of an interaction that took place among many entities constrained in specific configurations which are relational.Within this metaphor, the proposed method turns out to be a global topological application of the S[B] paradigm for modeling complex systems.

Affiliation:
School of Science and Technology, University of Camerino, Camerino, Italy.

ABSTRACT

In order to define a new method for analyzing the immune system within the realm of Big Data, we bear on the metaphor provided by an extension of Parisi's model, based on a mean field approach. The novelty is the multilinearity of the couplings in the configurational variables. This peculiarity allows us to compare the partition function [Formula: see text] with a particular functor of topological field theory-the generating function of the Betti numbers of the state manifold of the system-which contains the same global information of the system configurations and of the data set representing them. The comparison between the Betti numbers of the model and the real Betti numbers obtained from the topological analysis of phenomenological data, is expected to discover hidden n-ary relations among idiotypes and anti-idiotypes. The data topological analysis will select global features, reducible neither to a mere subgraph nor to a metric or vector space. How the immune system reacts, how it evolves, how it responds to stimuli is the result of an interaction that took place among many entities constrained in specific configurations which are relational. Within this metaphor, the proposed method turns out to be a global topological application of the S[B] paradigm for modeling complex systems.